Bayesian Inference with M-splines on Spectral Measure of Bivariate Extremes

Abstract

We consider a Bayesian methodology with M-splines for the spectral measure of a bivariate extreme-value distribution. The tail of a bivariate distribution function F in the max-domain of attraction of an extreme-value distribution function G may be approximated by that of its extreme value attractor. The function G is characterized by a probability measure with expectation equal to 1/2, called the spectral measure, and two extreme-value indices. This spectral measure determines the tail dependence structure of F. The approximation of the spectral measure is proposed thanks to a non-parametric Bayesian estimator that guarantees to fulfill a moment and a shape constraint. The problem of routine calculation of posterior distributions for both coefficients and knots of M-splines is addressed using the Markov chain Monte Carlo (MCMC) simulation technique of reversible jumps.